Annals of Mathematics, 152 (2000), 331–367
Curvature and symmetry
of Milnor spheres
By Karsten Grove and Wolfgang Ziller*
Dedicated to Detlef Gromoll on his 60th birthday
Introduction
Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing
problems in Riemannian geometry has been whether there are exotic spheres
with positive curvature. It is well known that there are exotic spheres that do
not even admit metrics with positive scalar curvature [Hi]. On the other hand,
there are many examples of exotic spheres with positive Ricci curvature (cf.
[Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where
it is shown that every exotic sphere that bounds a parallelizable manifold
has a metric of positive Ricci curvature. This includes all exotic spheres in
dimension 7. So far, however, no example of an exotic sphere with positive
sectional curvature has been found. In fact, until now, only one example of an
exotic sphere with nonnegative sectional curvature was known, the so-called
Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we
prove:
Theorem A. Ten of the 14 exotic spheres in dimension 7 admit metrics
of nonnegative sectional curvature.
In this formulation we have used the fact that in the Kervaire-Milnor
group, Z28 = Diff + (S 6 )/Diff + (D7 ), of oriented diffeomorphism types of homotopy 7-spheres, a change of orientation corresponds to the inverse; hence
the numbers 1 to 14 correspond to the distinct diffeomorphism types of exotic
7-spheres.
The exotic spheres that occur in this theorem are exactly those that can be
exhibited as 3-sphere bundles over the 4-sphere, the so-called Milnor spheres.
Each such exotic sphere can be written as an S 3 bundle in infinitely many
distinct ways; cf. [EK]. Our metrics are submersion metrics on these sphere
∗ The
first named author was supported in part by the Danish National Research Council and
both authors were supported by a RIP (Research in Pairs) from the Forschungsinstitut Oberwolfach
and by grants from the National Science Foundation.
332
KARSTEN GROVE AND WOLFGANG ZILLER
bundles and we will obtain infinitely many nonisometric metrics on each of
these exotic spheres; see Proposition 4.8. We do not know if any of the four
remaining exotic spheres in dimension 7 admit metrics with nonnegative curvature, or if the metrics on the Milnor spheres above can be deformed to positive
curvature. But no obstructions are known either.
Another central question in Riemannian geometry is, to what extent the
converse to the celebrated Cheeger-Gromoll soul theorem holds [CG]. The soul
theorem implies that every complete, noncompact manifold with nonnegative
sectional curvature is diffeomorphic to a vector bundle over a compact manifold with nonnegative curvature. The converse is the question which total
spaces of vector bundles over compact nonnegatively curved manifolds admit
(complete) metrics with nonnegative curvature. In one extreme case, where
the base manifold is a flat torus, there are counterexamples [OW], [Ta]. In another extreme case, where the base manifold is a sphere (the original question
asked by Cheeger and Gromoll) no counterexamples are known. But there are
also very few known examples, all of them coming from vector bundles whose
principal bundles are Lie groups or homogeneous spaces (cf. [CG], [GM], [Ri1],
and [Ri2]). It is easy to see that the total space of any vector bundle over
S n with n ≤ 3 admits a complete nonnegatively curved metric. (For n = 5,
see Proposition 3.14.) Another one of our main results addresses the first
nontrivial case.
Theorem B. The total space of every vector bundle and every sphere
bundle over S 4 admits a complete metric of nonnegative sectional curvature.
The special case of S 2 bundles over S 4 will give rise to infinitely many
nonnegatively curved 6-manifolds with the same homology groups as CP 3 , but
whose cohomology rings are all distinct; see (3.9).
From a purely topological relationship between bundles with base S 4 and
7
S (cf. Section 3 and [Ri3]) it will follow that most of the vector bundles and
sphere bundles over S 7 admit a complete metric of nonnegative curvature; see
(3.13). In [GZ2] we will use the constructions of this paper to also analyze
bundles with base CP 2 , CP 2 # ± CP 2 , and S 2 × S 2 .
From representation theory it is well known that any linear action of the
rotation group SO(3) has points whose isotropy group contains SO(2). A proof
of this assertion for general smooth actions of SO(3) on spheres was offered
in [MS]. However, this turned out to be false. In fact, among other things,
Oliver [Ol] was able to construct a smooth SO(3) action on the 8-disc D8 ,
whose restriction to the boundary 7-sphere S 7 is almost free, i.e., has only
finite isotropy groups. Explicitly, the isotropy groups of the example in [Ol]
are equal to 1, Z2 , D2 , D3 and D4 . By completely different methods we exhibit
infinitely many such actions on the 7-sphere.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
333
Theorem C (Exotic symmetries of the Hopf fibration). For each n ≥ 1
there exists an almost free action of SO(3) on S 7 which preserves the Hopf
fibration S 7 → S 4 and whose only isotropy groups, besides the principal isotropy
group 1, are the dihedral groups D1 = Z2 , D2 = Z2 × Z2 , Dn , Dn+1 , Dn+2 ,
and Dn+3 . Furthermore, these actions do not extend to the disc D8 if n ≥ 4.
In the case of the exotic 7-spheres we produce the first such examples.
Theorem D. Let Σ7 be any (exotic) Milnor sphere. Then there exist
infinitely many inequivalent almost free actions of SO(3) on Σ7 , one or more
for each fibration of Σ7 by 3-spheres, preserving this fibration.
Since the SO(3) actions in Theorems C and D take fibers to fibers, they
induce an action of SO(3) on the base S 4 . This action of SO(3) on S 4 is a fixed
action, which yields the well-known decomposition of S 4 into isoparametric
hypersurfaces [Ca], [HL]. Hence our actions on S 7 and Σ7 can be viewed as
lifts of this action of SO(3) on S 4 to the total space of the S 3 fibrations.
All of the above results follow from investigations and constructions related to manifolds of cohomogeneity one, i.e., manifolds with group actions
whose orbit spaces are 1-dimensional. For closed manifolds this means that
the orbit space is either a circle (and all orbits are principal) or an interval. In
the first case it is easy to see that the manifold supports an invariant metric
with nonnegative curvature. In the second more interesting case, the interior
points of the interval correspond to principal orbits and the endpoints to nonprincipal orbits. Although very difficult, it is tempting to make the following
Conjecture. Any cohomogeneity one manifold supports an invariant
metric of nonnegative sectional curvature.
If true, this would imply in particular that the Kervaire spheres in dimension 4n + 1, which carry cohomogeneity one actions by SO(2)SO(2n + 1)
(see [HH]), and are exotic spheres if n is even, support invariant metrics of
nonnegative curvature. In [BH] it was shown that the singular orbits of the
cohomogeneity one actions on the Kervaire spheres have codimension 2 and
2n, and that they do not admit a metric with positive sectional curvature,
invariant under the group action, when n > 1.
One of our key results is a small step in the direction of this conjecture.
Theorem E. Any cohomogeneity one manifold with codimension two
singular orbits admits a nonnegatively curved invariant metric.
The importance of Theorem E is due to the surprising fact that the class
of cohomogeneity one manifolds with singular orbits of codimension two is
extremely rich. This is illustrated by our other key result.
334
KARSTEN GROVE AND WOLFGANG ZILLER
Theorem F. Every principal L bundle over S 4 with L = SO(3) or
SO(4) supports a cohomogeneity one L × SO(3) structure with singular orbits
of codimension 2.
Theorem B is now an easy consequence of Theorems E and F in conjunction with the Gray-O’Neill curvature formula for submersions. Theorem
A follows from B together with the diffeomorphism classification in [EK]; see
the discussion following Remark 4.6. The SO(3) actions in Theorems C and
D arise from this construction as well, since the group SO(3) commutes with
the principal bundle action and hence induces an action on every associated
bundle.
Another consequence of Theorem E is the following (see the discussion
after 2.8):
Theorem G. On each of the four (oriented ) diffeomorphism types homotopy equivalent to RP 5 there exist infinitely many nonisometric metrics with
nonnegative sectional curvature.
The existence of infinitely many cohomogeneity one actions on S 5 inducing
corresponding actions on each of the homotopy RP 5 ’s was first discovered by
E. Calabi (unpublished, cf. [HH, p. 368]), who explained this to us in 1994.
These actions can be viewed as the special case n = 1 of the Kervaire sphere
actions eluded to above. Among these they are the only ones where both
singular orbits have codimension 2, so that Theorem E can be evoked directly.
Using the same methods as in [Se], one shows that there do not exist any
SO(2)SO(3) invariant metrics with positive curvature on these 5-dimensional
cohomogeneity one manifolds. This implies that if we apply Hamilton’s flow
to our metrics of nonnegative curvature, one cannot obtain a metric of positive
curvature since Hamilton’s flow preserves isometries.
The paper is organized as follows. Section 1 is devoted to general properties of cohomogeneity one manifolds and to an important construction of
principal bundles in this framework. In Section 2 we prove Theorems E and
G. The constructions in Section 1 are used to prove Theorem F in Section 3.
In Section 4 we analyze induced SO(3) actions on associated bundles and derive Theorems C and D. Finally, in Section 5 we examine the geometry of our
examples in more detail and raise some open questions.
It is our pleasure to thank J. Shaneson for general help concerning topological questions, and R. Oliver for sharing his insight about SO(3)-actions on
discs. We would also like to acknowledge that after seeing a first version of
our manuscript in which we had forgotten to include the Calabi examples, H.
Rubinstein informed us that O. Dearricott had noticed that Theorem E would
yield the existence of metrics of nonnegative curvature on the exotic RP 5 ’s.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
335
1. Principal bundles and cohomogeneity one manifolds
We first recall some basic facts about manifolds of cohomogeneity one and
establish some notation.
Let M be a closed, connected smooth manifold with a smooth action of a
compact Lie group G. We say that the action G × M → M is of cohomogeneity
one if the orbit space M/G is 1-dimensional. A cohomogeneity one manifold
is a manifold with an action of cohomogeneity one.
Consider the quotient map π : M → M/G. When M/G is 1-dimensional,
it is either a circle S 1 , or an interval I. In the first case all G orbits are principal
and π is a bundle map. It then follows from the homotopy sequence of this
bundle that the fundamental group π1 (M ) of M is infinite. In the second case
there are precisely two nonprincipal G-orbits corresponding to the endpoints
of I, and M is decomposed as the union of two tubular neighborhoods of
the nonprincipal orbits, with common boundary a principal orbit. All of this
actually holds in the topological category (cf. [Mo]).
In the remaining part of this paper we will only consider the most interesting case, where M/G = I. For this we will make the description above more
explicit in terms of an arbitrary but fixed G-invariant Riemannian metric on
M , normalized so that with the induced metric, M/G = [−1, 1]. Fix a point
x0 ∈ π −1 (0) and let c : [−1, 1] → M be the unique minimal geodesic with
c(0) = x0 and π ◦ c = id[−1,1] . Note that c : R → M intersects all orbits orthogonally, and c : [2n − 1, 2n + 1] → M , n ∈ Z are minimal geodesics between the
two nonprincipal orbits, B± = π −1 (±1) = G · x± , x± = c(±1). Let K± = Gx±
be the isotropy groups at x± and H = Gx0 = Gc(t) , −1 < t < 1, the principal
isotropy group. By the slice theorem, we have the following description of the
tubular neighborhoods D(B− ) = π −1 ([−1, 0]) and D(B+ ) = π −1 ([0, 1]) of the
nonprincipal orbits B± = G/K± :
(1.1)
D(B± ) = G ×K± D`± +1
where D`± +1 is the normal (unit) disk to B± at x± . Hence we have the
decomposition
(1.2)
M = D(B− ) ∪E D(B+ )
where E = π −1 (0) = G·x0 = G/H is canonically identified with the boundaries
∂D(B± ) = G ×K± S `± , via the maps G → G × S `± , g → (g, ∓ċ(±1)). Note
also that ∂D`± +1 = S `± = K± /H. All in all we see that we can recover M
from G and the subgroups H and K± . In fact, two manifolds which carry
a cohomogeneity one action by G with the same isotropy groups H and K± ,
along a minimal geodesic between nonprincipal orbits, must be G-equivariantly
diffeomorphic.
336
KARSTEN GROVE AND WOLFGANG ZILLER
In general, suppose G is a compact Lie group and H ⊂ K± ⊂ G are
closed subgroups such that K± /H = S `± are spheres. It is well known
(cf. [Bes, p. 95]) that a transitive action of a compact Lie group K on a sphere
S ` is linear and is determined by its isotropy group H ⊂ K. Thus the diagram
of inclusions
(1.3)
determines a manifold
(1.4)
M = G ×K− D`− +1 ∪G/H G ×K+ D`+ +1
on which G acts by cohomogeneity one via the standard G action on G ×K±
D`± +1 in the first coordinate. Thus the diagram (1.3) defines a cohomogeneity
one manifold, and we will refer to it as a cohomogeneity one group diagram,
which we sometimes denote by H ⊂ {K− , K+ } ⊂ G. We also denote the
common homomorphism j+ ◦ i+ = j− ◦ i− by j0 : H → G.
We are now ready for the main construction in this section: Principal
bundles over cohomogeneity one manifolds.
Let L be any compact Lie group, and M any cohomogeneity one manifold
with group diagram H ⊂ {K− , K+ } ⊂ G. It is important to allow the G-action
on M to be noneffective, i.e. G and H have a common normal subgroup, since
this will produce more principal bundles over M ; see, e.g., (3.1), and (3.2).
For any Lie group homomorphisms φ± : K± → L, φ0 : H → L with
φ+ ◦ i+ = φ− ◦ i− = φ0 , let P be the cohomogeneity one L × G-manifold with
diagram
(1.5)
Clearly the subaction of L × G by L = L × {e} on P is free since
L ∩ (l, g)K± (l, g)−1 = (l, g)(L × {e} ∩ K± )(l, g)−1
as well as
L ∩ (l, g)H(l, g)−1 = (l, g)(L × {e} ∩ H)(l, g)−1
CURVATURE AND SYMMETRY OF MILNOR SPHERES
337
is the trivial group for all (l, g) ∈ L × G. Moreover, P/L = M since it has a
cohomogeneity one description H ⊂ {K− , K+ } ⊂ G. It is also apparent that
the nonprincipal orbits in P have the same codimension as the nonprincipal
orbits in M , as well as the same slice representation, since the normal bundles in
M pull back to the normal bundles in P under the principal bundle projection
P → M . In summary:
Proposition 1.6. For every cohomogeneity one manifold M as in (1.3)
and every choice of homomorphisms φ± : K± → L with φ+ ◦ i+ = φ− ◦ i− , the
diagram (1.5) defines a principal L bundle over M .
Note, moreover, that the L × G-action on P may well be effective even if
the G-action on M is not.
We now move on to discuss induced actions on associated bundles:
Let F be a smooth manifold on which L acts, L × F → F . Consider the
total space of the associated bundle V = P ×L F . Observe that the product
of the trivial G-action on F with the sub-action of G = {e} × G ⊂ L × G on
P induces a natural G-action on V .
Lemma 1.7 (Isotropy Lemma). The natural G-action on V = P ×L F
has exactly the following types of isotropy groups
φ−1
± (Lu )
and
φ−1
0 (Lu )
where Lu , u ∈ F are the isotropy groups of L × F → F .
Proof. Consider the L-orbit, L(x, u) = {(`x, `u) | ` ∈ L} of a point
(x, u) ∈ P × F . Then
GL(x,u)
= {g ∈ G | gL(x, u) = L(gx, u) = L(x, u)}
= {g ∈ G | ∃ ` ∈ L : (gx, u) = (`x, `u)}
= {g ∈ G | ∃ ` ∈ Lu : (`−1 , g) ∈ (L × G)x }.
ˆ ĝ)-conjugate of one of (φ+ , j+ )(K+ ), (φ− , j− )(K− )
However, (L×G)x is some (`,
or (φ0 , j0 )(H), and the claim follows.
2. Nonnegative curvature on homogeneous bundles
The purpose of this section is to prove Theorems E and G of the introduction.
As in [Ch1] we will construct nonnegative curvature metrics on M =
D(B− ) ∪E D(B+ ) (cf. 1.2) with the additional property that the common
boundary E = ∂D(B− ) = ∂D(B+ ) is totally geodesic in M . This is a very
strong restriction, which, by the soul theorem [CG], implies that also B± are to-
338
KARSTEN GROVE AND WOLFGANG ZILLER
tally geodesic. With this in mind, all we have to do is to construct G-invariant
nonnegative curvature metrics on the bundles D(B± ) = G ×K ± D`± +1
(cf. (1.1)), that agree on the common boundary E = G/H = G ×K ± S `±
and are product metrics near the boundary.
From the Gray-O’Neill curvature submersion formula (cf. [ON] or [Gr]),
we know that the product metric of a left invariant, Ad (K)-invariant metric of
nonnegative curvature on G with a K-invariant nonnegative curvature metric
on D`+1 (which is product near S ` = ∂D`+1 ) induces a G-invariant nonnegative
curvature metric on the quotient G ×K D`+1 (which is product near G/H =
G×K S ` = ∂(G×K D`+1 )). The difficulty in the above strategy is therefore, that
in general the restriction of such metrics on G×K− D`− +1 and on G×K+ D`+ +1
to G/H = G ×K− S `− = G ×K+ S `+ are different.
Consider any closed Lie subgroups H ⊂ K ⊂ G of a compact Lie group
G, with Lie algebras h ⊂ k ⊂ g. Fix any left invariant, Ad (K)-invariant
Riemannian metric, h , i on G and let m = k⊥ and p = h⊥ ∩ k relative to
this metric. On G/H and K/H we get induced (submersed) G-, respectively
K-invariant metrics which are also denoted by h , i. As usual we make the
identifications p + m ' TH G/H and p ' TH K/H via action fields; i.e., X +A →
∗ respectively.
(X + A)∗H and X → XH
The homogeneous space G/H can be identified with the orbit space G ×K
K/H of G×K/H by the K-action (k, (g, k̄H)) → (gk −1 , k k̄H). The
√identification is given by gH → K(g, H) with inverse K(g, kH) → gkH. By λK/H we
mean K/H endowed with the metric λh , i, where λ > 0. In this terminology
we have:
Lemma 2.1. The√G-invariant metric h , iλ on G/H induced from the
product metric on G × λK/H via G ×K K/H ' G/H is determined by
h , iλ|m = h , i|m
and
h , iλ|p =
λ
λ+1 h
, i|p .
Proof. The vertical space (= tangent space to K-orbit) at (1, H) ∈ G ×
K/H is given by
v
∗
T(1,H)
= h × {0} + {(−X, XH
) | X ∈ p}.
Thus (U, YH∗ ) ∈ T(1,H) G × K/H, U ∈ g, Y ∈ p is horizontal if and only if
U = Z + A ∈ p + m = h⊥ satisfies −hZ, Xi + λhY, Xi = 0 for all X ∈ p; i.e.
h
T(1,H)
= m × {0} + {(λY, YH∗ ) | Y ∈ p}.
Now (A, 0) projects to A ∈ m ⊂ TH G/H and (λY, YH∗ ) projects to (λ + 1)Y
∈ p ⊂ TH G/H. In particular, the horizontal lift of A ∈ TH G/H to (1, H) is
1
(λY, YH∗ ). This proves the claim since
(A, 0), and Y ∈ p ⊂ TH G/H lifts to λ+1
1
the norms of these vectors are given by k(A, 0)k2 = kAk2 and k λ+1
(λY, YH∗ )k2 =
1 2 2
λ
2
2
2
( λ+1 ) (λ kY k + λkY k ) = λ+1 kY k .
339
CURVATURE AND SYMMETRY OF MILNOR SPHERES
As an immediate consequence of this lemma, we see that if Q is a fixed
bi-invariant metric on G, and we choose h , i above as
h , i|m = Q|m
h , i|k = λ+1
λ Q|k
√
then the metric on G/H induced via G ×K λK/H as above, is the same as
the one induced directly via Q. This is essentially the method that Cheeger
used in [Ch1] to construct a nonnegatively curved metric on the connected
sum of two projective spaces. The problem now, however, is that in general
a metric like (2.2) on G has some negative sectional curvature, as we will see,
since a = λ+1
λ > 1.
We need to work in a slightly more general context. As before G is a
compact Lie group and k ⊂ g a subalgebra. Let K ⊂ G be the (immersed) Lie
subgroup of G with Lie algebra k; i.e. K need not be compact. As before let
Q be a fixed bi-invariant metric on g and a > 0. Define
(2.2)
Qa|m = Q|m
(2.3)
and
and
Qa|k = aQ|k
and denote again by Qa also the corresponding left and Ad (K) invariant metric
on G. We need the following curvature formulas for this left invariant metric
(see, e.g., [Es] and [DZ] for special cases).
Proposition 2.4. For any a > 0 let Ra be the curvature tensor of the
metric Qa defined in (2.3). Then for any A, B ∈ m and X, Y ∈ k we have
Qa (Ra (A + X, B + Y )(B + Y ), A + X)
=
1
4
°
°
°2
°
k[A, B]m + a[X, B] + a[A, Y ]k2Q + 14 °[A, B]k + a2 [X, Y ]°
Q
+ 14 a(1 − a)3 k[X, Y ]k2Q + 34 (1 − a) k[A, B]k + a[X, Y ]k2Q
where subscripts denote components. In particular, (G, Qa ) has nonnegative
curvature whenever 0 < a ≤ 1, or if k is abelian and a ≤ 43 .
Proof. For a = 1 this is the well-known formula for the sectional curvature
of a bi-invariant metric. For a 6= 1, we claim that Qa is a submersed metric.
Indeed, on G × K consider the bi-invariant (semi-) Riemannian metric induced
a
we claim
from h , i = Q × bQ|k (b negative allowed) on g × k. When b = 1−a
that the map G × K → G, (g, k) 7→ gk is a (semi-) Riemannian submersion.
In fact this can be viewed as a special case of (2.1) above, when H is trivial,
by noticing that in this case the vertical space given by
v
T(1,1)
= {(−X, X) | X ∈ k} ⊂ T(1,1) G × K
is nondegenerate since b 6= −1. (This would not be true in the general case
where h 6= {0}.) The rest of the argument in (2.1) carries over verbatim and
b
= a in the k-direction.
we see that the submersed metric on G is scaled by b+1
340
KARSTEN GROVE AND WOLFGANG ZILLER
To compute the sectional curvature, we use the Gray-O’Neill formula.
Consider a 2-plane in T1 G = g spanned by A + X and B + Y as in (2.4).
b
1
X, b+1
X) and
The corresponding horizontal lifts to T(1,1) G × K are (A + b+1
b
1
(B + b+1 Y, b+1 Y ), respectively. Moreover, when extending the G-coordinate
to left-invariant vector fields and the K-coordinate to right-invariant vector
fields, the resulting fields are easily seen to be horizontal. The Gray-O’Neill
formula then yields:
Qa (Ra (A + X, B + Y )(B + Y ), A + X) = α + 34 β,
where
³
= hRG×K (A +
α
(B +
and
β
°h
°
° (A +
=
Now
α
=
=
°
°
°[A +
1
4
°
°
°[A, B]m +
1
4
b
1
b+1 Y, b+1 Y
), (A +
b
1
b+1 X, b+1 X), (B
1
4
+
b
1
b+1 X, b+1 X), (B
b
b+1 X, B
+
b
b+1 Y
+
+
b
1
b+1 Y, b+1 Y
b
1
b+1 X, b+1 X)ig×k
b
1
b+1 Y, b+1 Y
°2
°
iv °2
°
) °
g×k
°
°
1
4
=
.
°2
°
1
1
]° + 14 b °[ b+1
X, b+1
Y ]°
Q
Q
°2
°
b
([X,
B]
+
[A,
Y
])
°
b+1
Q
°
°2
°
°
b 2
1 4
) [X, Y ]° + 14 b( b+1
) k[X, Y ]k2Q
°[A, B]k + ( b+1
Q
where we have used [m, k] ⊂ m and [k, k] ⊂ k. In terms of a =
1
a
1 − a = b+1
and b = 1−a
) we have
α
´
)
b
b+1
(and hence
°
°
°2
°
k[A, B]m + a[X, B] + a[A, Y ]k2Q + 14 °[A, B]k + a2 [X, Y ]°
+
Q
1
4 a(1
− a) k[X, Y
3
]k2Q .
Using the fact that for right-invariant vector fields X ∗ , Y ∗ , [X ∗ , Y ∗ ] = −[∗ X,∗ Y ]
= −[X, Y ] in terms of left-invariant vector fields, we get:
β
°³
°
= ° [A, B]m +
b
b+1 [A, Y
]+
b
b+1 [X, B]
+ [A, B]k
´v °2
°
b 2
1 2
+( b+1
) [X, Y ], −( b+1
) [X, Y ] °
g×k
=
since
m × 0 ⊂ T h.
°³
´v °2
°
°
b 2
1 2
) [X, Y ], −( b+1
) [X, Y ] °
° [A, B]k + ( b+1
g×k
,
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CURVATURE AND SYMMETRY OF MILNOR SPHERES
For U, V ∈ k we have
(U, V ) =
1
b+1 (b(U
and hence (U, V )v =
+ V ), U + V ) +
1
b+1 (U
n
1
b+1 (−(−U
+ bV ), −U + bV )
− bV, −U + bV ). Moreover,
o
1 2
k(U, V )v k2g×k = ( b+1
) kU − bV k2Q + bk − U + bV k2Q =
This yields
β
=
=
1
b+1
1
b+1 kU
− bV k2Q .
°
°2
°
°
b 2
1 2
) [X, Y ] + b( b+1
) [X, Y ]°
°[A, B]k + ( b+1
(1 − a) k[A, B]k + a[X, Y
Q
]k2Q
which completes the proof of the curvature formula.
If a ≤ 1 all terms are nonnegative. For a > 1 the latter two terms will be
negative in general. However, if k is abelian the formula reduces to
Qa (Ra (A + X, B + Y )(B + Y ), A + X)
=
1
4
k[A, B]m + a[X, B] + a[A, Y ]k2Q + (1 − 34 a) k[A, B]k k2Q
which is nonnegative for a ≤ 4/3 as claimed.
Remark 2.5. In general there are two planes with strictly negative curvature on (G, Qa ) for any a > 1 arbitrarily close to 1. Indeed, one can usually
easily find two planes spanned by A+X and B+Y with [A, B] = −a2 [X, Y ] and
[X, B] + [A, Y ] = 0 which will have negative sectional curvature if [X, Y ] 6= 0.
We are now ready to prove the main result of this section.
Theorem 2.6. Suppose G is a connected, compact Lie group and H ⊂
K ⊂ G are closed subgroups with K/H = S 1 . Then for any bi -invariant metric
on G, there is a G-invariant nonnegatively curved metric on G ×K D2 which is
a product near the boundary G/H = G ×K S 1 , and so that the metric restricted
to G/H is induced from the given bi-invariant metric on G.
Proof. Fix a bi-invariant metric Q on g and let m = k⊥ , p = h⊥ ∩ k as
before. By assumption, p is 1-dimensional and is hence an abelian subalgebra
of g. Moreover, if H̄ ⊂ H is the ineffective kernel of the K-action on S 1 =
K/H = (K/H̄)/(H/H̄) we have h̄ = h since the isotropy group of an effective
action on S 1 is finite. Since H̄ is normal in K, h and hence p is preserved by
Ad (K). This implies that the metric Q̄a on G defined by
Q̄a|m = Q|m , Q̄a|p = aQ|p
and
Q̄a|h = Q|h
is Ad (K)-invariant. Since p is a subalgebra, this metric can also be viewed
as in (2.3) with k = p and K = exp(p) since in (2.3) we allowed K to be
a noncompact Lie group. Hence (2.4) implies that the metric Q̄a on G has
nonnegative sectional curvature if a ≤ 4/3.
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KARSTEN GROVE AND WOLFGANG ZILLER
Let K/H = S 1 be equipped with the metric induced from Q̄a| . By
k
√
Lemma (2.1), the metric on G ×K λS 1 ' G/H is then given by Q on m and
λ
λ+1 aQ on p. Now pick, e.g., a = 4/3, λ = 3 and a K-invariant metric on
near the boundary circle
D2 with nonnegative curvature, which
√ is a product
√
∂D2 = S 1 , and on S 1 is the metric 3S 1 = 3K/H from above.
The quotient metric on G ×K D2 induced from the product metric on G
and on D2 has all the desired properties claimed in (2.6) .
From (2.6) and the discussion in the beginning of this section, it follows
immediately that we can construct nonnegatively curved metrics on each half
G ×K± D2 , matching smoothly near ∂(G ×K− D2 ) ' G/H ' ∂(G ×K+ D2 ) to
yield G-invariant metrics on M = G ×K− D2 ∪E G ×K+ D2 with nonnegative
curvature. This finishes the proof of Theorem E.
Remark 2.7. For a metric on D2 we can choose a rotationally symmetric
metric dt2 +f (t)2 dθ2 , where f is a concave function which is odd with f 0 (0) = 1
in order to guarantee smoothness of the metric. Suppose K/H = (S 1 , Q) is a
circle of length 2πr. Then the induced metric on the principal orbit G/H at
c(t) (where t = 0 corresponds to the singular orbit G/K) can be described as
2a
(t)
G ×K rf√
K/H which, using (2.1), is then given by Q on m and f 2f+ar
2 Q on p.
a
2
ar
Hence we need to choose a t0 such that f 2 (t) = a−1
, for t ≥ t0 . Notice that
the larger the radius r is, or if we choose 1 < a ≤ 4/3 close to 1, the larger t0
needs to be, and hence the diameter of M will be large.
Remark 2.8. In the case where a nonregular orbit is exceptional, i.e., is a
hypersurface, one can just choose the bi-invariant metric on G itself to induce
a metric on the disc bundle G ×K D1 , which then has the same properties as
in Theorem 2.6. Hence one obtains a nonnegatively curved metric on every
cohomogeneity one manifold with nonregular orbits of codimension ≤ 2.
We point out that there are many cohomogeneity one manifolds with
nonnegative curvature, whose singular orbits have codimension bigger than
2. One large class is the linear cohomogeneity one actions on round spheres
S n (1), classified in [HL], and characterized as the isotropy representations of
compact rank two symmetric spaces. There are also many isometric cohomogeneity one actions on compact symmetric spaces with their natural metric of
nonnegative curvature, recently classified in [Ko] in the irreducible case. In
almost all of these examples, none of the principal orbits are totally geodesic.
The difficulty in proving the conjecture that every cohomogeneity one manifold carries a metric with nonnegative curvature may lie in that one needs a
better understanding of how to glue the two halves together without making
the middle totally geodesic.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
343
One particularly intriguing class of cohomogeneity one manifolds are the
2n − 1-dimensional Brieskorn varieties defined by the equations
z0d + z12 + · · · zn2 = 0 ,
|z0 |2 + · · · |zn |2 = 1.
For n odd and d odd, they are homeomorphic to spheres, and if in addition
d ≡ ±1 mod 8, they are diffeomorphic to spheres, whereas for d ≡ ±3 mod 8,
they are diffeomorphic to the Kervaire sphere. The Kervaire sphere is an exotic
sphere if 2n − 1 ≡ 1 mod 8 or more generally if n + 1 is not a power of 2. As
discovered in [HH], the Brieskorn variety carries a cohomogeneity one action by
SO(2)SO(n) defined by (eiθ , A)(z0 , · · · , zn ) = (e2iθ z0 , eidθ A(z1 , · · · , zn )t ). This
action was examined in detail in [BH], where they showed that the group
picture (for d odd) is given by K− = SO(2) × SO(n − 2), K+ = O(n − 1), H =
Z2 × SO(n − 2), with embeddings given by
(eiθ , A) ∈ K− ⊂ SO(2)SO(2)SO(n − 2) → (eiθ , R(dθ), A)
with R(dθ) a rotation by angle dθ, A ∈ K+ → (det(A), (det(A), A)), and (ε, A)
∈ Z2 × SO(n − 2) = H → (ε, (ε, ε, A)). In particular, one obtains a different
action for each odd d, and the nonprincipal orbits have codimension 2 and
n − 1.
In the special case n = 3 and d odd, where these actions define a cohomogeneity one action on S 5 , they were first discovered by E. Calabi, who also
observed that they descend to cohomogeneity one actions on the homotopy
projective spaces S 5 /Z2 , where Z2 is the element −id ∈ SO(2). In [Lo] it was
shown that this homotopy projective space contains four (oriented) diffeomorphism types, according to d ≡ 1, 3, 5, 7 mod 8, and two homeomorphism types,
according to d ≡ ±1, ±3 mod 8. Notice that it is not known if all of the exotic
RP 5 ’s admit any orientation-reversing diffeomorphisms. Hence it is conceivable
that two of the exotic differentiable structures are the same. In any case, each
of the possible differentiable structures on RP 5 carries infinitely many cohomogeneity one actions by SO(2)SO(3), and since the codimension of the singular
orbits in this case are both equal to two, they all admit an invariant metric with
nonnegative sectional curvature by Theorem E. One easily shows that the effective group picture is given by G = SO(2)SO(3), K− = SO(2) with embedding
eiθ → (e2iθ , (R(dθ), id)), K+ = O(2) with embedding A → (1, (det(A), A)) and
H = Z2 = h(1, diag (−1, −1, 1))i.
To finish the proof of Theorem G, we need to show that these metrics
are never isometric to each other. For this we first note that if the action of
SO(2)SO(3) extends to a transitive action, then it must be linear and hence
corresponds to the case d = 1 which is the well-known tensor product action. If d > 1, we will argue that SO(2)SO(3) is the identity component of
the isometry group, and since the group actions are never conjugate to each
other, the corresponding metrics cannot be isometric either. Notice that any
344
KARSTEN GROVE AND WOLFGANG ZILLER
isometries, besides the elements of SO(2)SO(3), must preserve the G orbits
and hence induce isometries of the homogeneous metrics on the principal orbits SO(2)(SO(3)/Z2 ). One easily shows that for any invariant metric on this
homogeneous space, any further isometries in the identity component come
from right translations by N SO(3) (Z2 )/Z2 . But these right translations do not
extend to G/K+ and hence are not well defined on M . This finishes the proof
of Theorem G.
3. Topology of principal bundles
In this section we discuss the proof of Theorem F from the introduction.
First note that over S 4 , every principal SO(2) bundle is trivial and well known
obstruction theory implies that every k-dimensional vector bundle with k > 4
is the direct sum of a 4-dimensional bundle and a trivial bundle. Hence we
only need to examine principal SO(3) and SO(4) bundles. This is also why
Theorems E and F, together with the Gray-O’Neill submersion formula, imply
Theorem B.
To employ the methods of Section 1 we begin by describing the well-known
cohomogeneity one action by SO(3) on S 4 in a language that will be needed
for our construction of principal bundles. Let
V = {A | A a 3 × 3 real matrix with A = At , tr (A) = 0}.
Then V is a 5-dimensional vector space with inner product hA, Bi = tr AB.
SO(3) acts on V via conjugation g · A = gAg −1 and this action preserves the
inner product and hence acts on S 4 (1) ⊂ V . Every point in S 4 (1) is conjugate
P
P
to a matrix in F = {diag (λ1 , λ2 , λ3 ) | λi = 0, λ2i = 1} and hence the quotient space is 1-dimensional. The singular orbits B± consist of those matrices A
with two eigenvalues λi the same, negative for B− and positive for B+ . Clearly,
4
all orbits
we can choose
F is a great circle
√ in S√ (1) that
√ is orthogonal to √
√ and √
x− = diag (2/ 6, −1/ 6, −1/ 6), x+ = diag (1/ 6, 1/ 6, −2/ 6) and hence
K− = S(O(1)O(2)), K+ = S(O(2)O(1)) ⊂ SO(3). As long as λ1 > λ2 > λ3 we
obtain the principal isotropy group H = S(O(1)O(1)O(1)) = Z2 × Z2 . Notice
that B− and B+ are both Veronese surfaces in S 4 (1) which are antipodal to
each other at distance π/3.
Next, we lift these groups into S 3 under the two-fold cover S 3 = Sp(1) →
SO(3) which sends q ∈ Sp(1) into a rotation in the 2-plane Im (q)⊥ ⊂ Im (H)
with angle 2θ, where θ is the angle between q and 1 in S 3 (1). After renumbering the coordinates, the group K− lifts to Pin (2) = {eiθ } ∪ {jeiθ } which we
abbreviate to eiθ ∪ jeiθ . Similarly, K+ lifts to Pin (2) = ejθ ∪ iejθ , and H =
S(O(1)O(1)O(1)) ⊂ SO(3) lifts to the quaternion group Q = {±1, ±i, ±j, ±k}.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
345
Thus the group diagram for S 4 is
(3.1)
We are now in a position to construct principal SO(3) bundles over S 4 . Since
the second Stiefel-Whitney class w2 of the principal bundle SO(3) → P ∗ → S 4
is zero, there exists a two-fold cover P of P ∗ such that P → S 4 is a principal
S 3 bundle. We first construct a cohomogeneity one action by G on P , with
S 3 ⊂ G, which then induces a cohomogeneity one action on P ∗ since P ∗ = P/σ,
with σ = −1 central in S 3 , as long as σ is also central in G.
Principal bundles S 3 → P → S 4 are classified by an element in π3 (S 3 )
= Z and hence by an integer k. Equivalently, we can consider the classifying
map of the bundle f : S 4 → BS 3 = HP ∞ and then k = f ∗ (x)[S 4 ] where x ∈
H 4 (HP ∞ , Z) = Z is the generator corresponding to HP 1 ⊂ HP ∞ . Hence we
can also consider k as the Euler class of the principal S 3 bundle, regarded as
a sphere bundle over S 4 , and evaluated on the fundamental class. Indeed the
latter follows from the fact that the universal principal S 3 bundle over HP ∞
is the Hopf bundle with Euler class x. Throughout the rest of the paper we
denote by Pk → S 4 the principal S 3 bundle with Euler class k.
We can now use the S 3 cohomogeneity one action on S 4 in (3.1) and the
main construction in (1.6) to arrive at the following group diagram:
(3.2)
where 4Q = {±(1, 1), ±(i, i), ±(j, j), ±(k, k)}. In order for H to be a subgroup
of K± , we need that p± ≡ 1 mod 4 and then we get K± /H = S 1 . Hence (3.2)
defines a cohomogeneity one manifold Pp− ,p+ . Notice that the action of S 3 ×S 3
is again ineffective, the effective version being S 3 × S 3 / ± (1, 1) = SO(4). As in
(1.6), it now follows that S 3 = S 3 × 1 acts freely on Pp− ,p+ and that P/S 3 is a
cohomogeneity one manifold as in (3.1) and hence equivariantly diffeomorphic
346
KARSTEN GROVE AND WOLFGANG ZILLER
to S 4 . Thus we obtain a principal bundle
S 3 → Pp− ,p+ → S 4 .
Since σ = (−1, 1) is central in S 3 × S 3 , we also obtain a cohomogeneity one
action by SO(3) × SO(3) on the principal SO(3) bundle P ∗ = P/(−1, 1) → S 4 .
To identify the principal bundle, we prove:
Proposition 3.3.
k = (p2− − p2+ )/8.
The principal S 3 bundle Pp− ,p+ → S 4 is classified by
Proof. The Gysin sequence of the sphere bundle S 3 → Pk → S 4 yields
that the nonzero cohomology groups of Pk are: H 0 = H 7 = Z and H 4 (Pk , Z) =
Z/|k|Z if k 6= 0 and H 3 = H 4 = Z if k = 0. Hence we can recognize |k| by
computing the cohomology groups of Pp− ,p+ .
To do this in general for a cohomogeneity one manifold M : H ⊂ {K− , K+ }
⊂ G, we use the Meyer-Vietoris sequence, where U± = D(B± ) = G ×K± D`± +1
deformation retracts to B± = G/K± and U− ∩ U+ = G/H. Hence we get a
long exact sequence
(3.4)
∗ −π ∗
π−
+
→ H i−1 (B− ) ⊕ H i−1 (B+ ) −−−−→ H i−1 (G/H) → H i (M )
→ H i (B− ) ⊕ H i (B+ ) →
where π± are the projections of the sphere bundles G/H = G ×K± S `± =
∂D(B± ) → B± = G/K± . Notice that in our case of (3.2) above, the restriction
of the principal S 3 bundle Pp− ,p+ → S 4 to the S 3 orbits S 3 /eiθ ∪ jeiθ ' RP 2 '
S 3 /ejθ ∪ iejθ and S 3 /Q in S 4 are all trivial, since the classifying space HP ∞
for principal S 3 bundles is 3-connected. Thus B± = G/K± = S 3 × RP 2
and G/H = S 3 × (S 3 /Q) up to diffeomorphism. In particular we obtain:
H 3 (B± , Z) = Z, H 4 (B± , Z) = 0, and H 3 (G/H, Z) = Z + Z, and the MeyerVietoris sequence (3.4) for P = Pp− ,p+ becomes:
H 3 (P ) → H 3 (B− ) ⊕ H 3 (B+ )
=
=
∗ −π ∗
π−
+
Z + Z −−−−→ H 3 (G/H)
Z + Z → H 4 (P ) → 0.
∗ − π ∗ . In
In order to compute H 4 (P ), we need to compute the cokernel of π−
+
∗
∗
our case, this cokernel is determined by the determinant of π− − π+ : Z2 → Z2 .
If the determinant is equal to 0, then H 4 (P ) = Z, and if it is nonzero then
H 4 (P ) is a cyclic group with order the absolute value of the determinant.
Consider the commutative diagram:
S3 × S3
(3.5)


η
y
τ±
o
−−−−→ S 3 × S 3 /K±


µ±
y
π±
S 3 × S 3 /H −−−−→ S 3 × S 3 /K±
347
CURVATURE AND SYMMETRY OF MILNOR SPHERES
o are the identity components of K . The maps µ are two-fold covers
where K±
±
±
o = S 3 × (S 3 /eiθ ) = S 3 × S 2 , and since
and as before it follows that S 3 × S 3 /K±
o ) is an
S 3 × S 3 /K± = S 3 × RP 2 , it follows that µ∗± : H 3 (G/K± ) → H 3 (G/K±
isomorphism. The map η is an 8-fold cover and if we write S 3 × S 3 /H = S 3 ×
(S 3 /Q), then η ∗ in H 3 is an isomorphism on the first factor and multiplication
by 8 on the second. Hence η ∗ : H 3 (S 3 × S 3 /H, Z) = Z2 → H 3 (S 3 × S 3 , Z) = Z2
∗ − π ∗ ) = det(τ ∗ − τ ∗ )/8. It remains to
has determinant 8 and therefore det(π−
+
−
+
determine the induced map in H 3 for the S 1 bundle τ± . For this purpose we
consider the following commutative diagram of fibrations, where we drop the
± index for the moment (see e.g. [WZ, p. 228]).
S1
(3.6)



y
τ± =τ
ρ1
−−−→ S 3 × S 3 −−−→ S 3 × S 3 /K o −−−→


id
y



y
h
S 1 × S 1 −−−→ S 3 × S 3 −−−→
S2 × S2
BS 1


r
y
ρ2
−−−→ BS 1 × BS 1
coming from the S 1 bundle τ and the S 1 × S 1 bundle h (product of Hopf
bundles). If we let H ∗ (BS 1 ) = Z[s] and H ∗ (BS 1 × BS 1 ) = Z[t1 , t2 ], then
r∗ (t1 ) = ps, r∗ (t2 ) = s since the inclusion S 1 → S 1 × S 1 is given by eiθ →
(eipθ , eiθ ). If we set H ∗ (S 3 × S 3 ) = Λ(u, v), then the only nonzero differentials
in the spectral sequence for ρ2 are d2 (u) = t21 , d2 (v) = t22 . By naturality the
differentials in the spectral sequence for ρ1 are given by d2 (u) = p2 s2 , d2 (v) = s2
and hence a generator 1 in H 3 (S 3 × S 3 /K o ) goes to (−u, p2 v) under τ ∗ . Thus
τ− (1) = (−u, p2− v), τ+ (1) = (−u, p2+ v) and the matrix of τ−∗ − τ+∗ is given by:
Ã
−1
1
2
2
p− −p+
!
which implies that |k| = |p2− − p2+ |/8.
Next we will show that k = ±(p2− − p2+ )/8 with a fixed choice of sign,
that is, the sign does not depend on p− , p+ . For this, consider the manifolds
Pp7− ,p+ and Pp7+ ,p− . We claim that the Euler class of the corresponding S 3
bundles differ by a sign. First note that the antipodal map −id: S 4 → S 4
interchanges the two halves of S 4 relative to the decomposition (3.1). Since
it is orientation-reversing the Euler class of the pullback bundle (−id)∗ Pp− ,p+
is the negative of Pp− ,p+ . Moreover, (−id)∗ Pp− ,p+ is a cohomogeneity one
manifold with diagram as for Pp− ,p+ , except the roles of i and j are switched.
Precomposing the S 3 × S 3 action by (A, A): S 3 × S 3 → S 3 × S 3 where A is the
inner automorphism of S 3 given by A(i) = j, A(j) = i and A(k) = k −1 we see
that Pp+ ,p− and (−id)∗ Pp− ,p+ are equivariantly diffeomorphic. In particular,
the Euler class of Pp+ ,p− and Pp− ,p+ have opposite signs.
To see which sign is the correct one (although this is not important for
our main results), we need to compute the Euler class in one particular case.
348
KARSTEN GROVE AND WOLFGANG ZILLER
For this one can take the well-known cohomogeneity one action by SO(4) on
ˆ 1◦ (the isotropy
S 7 (see e.g. [TT] ) which is given by the representation 3◦⊗
representation of the rank 2 symmetric space G2 /SO(4)). This action preserves
the Hopf fibration S 3 → S 7 → S 4 with Euler class 1. By computing the
isotropy groups of this SO(4) action, one shows (cf. [GZ1]) that they are the
same as the ones for P−3,1 and hence k = (p2− − p2+ )/8 as claimed.
Corollary 3.7. Every principal S 3 , respectively SO(3), bundle over S 4
has a cohomogeneity one action by G = SO(4), respectively G = SO(3)×SO(3),
in fact in general several inequivalent ones.
Proof. We only need to convince ourselves that every integer k can be
written as (p2− − p2+ )/8, where p± ≡ 1 mod 4. Set p− = 4r + 1, p+ = 4s + 1
and hence k = (r − s)(2r + 2s + 1). Then if we let r = −s, we get k = −2s,
for r = s + 1 we get k = 4s + 3 and for r = s − 1 we get k = −4s + 1.
These solutions can also be written in the following more convenient form:
(p− , p+ ) = (2k + 1, −2k + 1) if k ≡ 0 mod 2, (p− , p+ ) = (−k − 2, −k + 2) if
k ≡ 1 mod 4 and (p− , p+ ) = (k + 2, k − 2) if k ≡ 3 mod 4. Hence every integer
k can be achieved, in general in several different ways.
Remark 3.8. For each value of k 6= 0 there exist only finitely many
solutions of k = (p2− − p2+ )/8 = (r − s)(2r + 2s + 1), which can all be described
as follows: Set m = r − s and n = 2r + 2s + 1. Then k = nm with n odd, and
r = (2m + n − 1)/4, s = (−2m + n − 1)/4. Hence for each way of writing k as
a product nm with n odd (including sign changes for both n and m), we get
a solution for p− and p+ , if r and s are integers. Notice that if k = 2t , then
m = 2t , n = 1; hence one only gets one solution: p− = 2k +1, p+ = −2k +1 and
it is not hard to see that in all other cases, one obtains several solutions. Thus
all principal S 3 bundles with k 6= 2t have several inequivalent cohomogeneity
one actions by G = SO(4). If, for example, k = 105, then the following is
the complete set of eight solutions: (p− , p+ ) = (29, 1), (−31, −11), (37, −23),
(41, 29), (−47, 37), (73, −67), (−107, −103), (−211, 209).
If k = 0, i.e., on P 7 = S 4 × S 3 , we obtain infinitely many inequivalent
cohomogeneity one actions corresponding to p− = p+ .
Using the principal S 3 bundle Pk with Euler class k, we can consider the
associated 2-sphere bundle Mk = Pk ×S 3 S 2 → S 4 , where S 3 acts on S 2 via the
two-fold cover S 3 → SO(3). This can also be described as Mk = Pk /S 1 with
S 1 ⊂ S 3 . We now observe the following interesting consequence of our results:
Corollary 3.9. The total space of the S 2 bundles Mk → S 4 , which
admit a metric with nonnegative sectional curvature, have the same integral
cohomology groups as CP 3 , but distinct cohomology rings for k ≥ 2.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
349
Proof. The Gysin sequence of the sphere bundle S 2 → Mk → S 4 yields
that the nonzero cohomology groups H ∗ (Mk , Z) are H 0 = H 2 = H 4 = H 6 = Z.
From the Gysin sequence of the circle bundle S 1 → Pk → Mk , and H 4 (Pk , Z)
= Zk , we get that if x, y are the generators in H 2 (Mk , Z) and H 4 (Mk , Z), then
x2 = ky. Hence Mk all have the same cohomology groups as CP 3 , but distinct
cohomology rings, as long as k ≥ 2. Notice that Mk and M−k are diffeomorphic,
M±1 is diffeomorphic to CP 3 , and M0 is diffeomorphic to S 2 × S 4 .
Next, we consider the case of principal S 3 × S 3 bundles P over S 4 and the
corresponding principal SO(4) bundles P ∗ → S 4 with P ∗ = P/(−1, −1). These
bundles are classified by elements of π3 (S 3 × S 3 ) = π3 (SO(4)) = Z ⊕ Z and
hence by pairs of integers (k, l). For this identification, we use the convention
in [Mi]: To an element (k, l) we associate the element in π3 (SO(4)) given by
q ∈ S 3 → (u → q k uq l ) ∈ SO(4). Under the two-fold cover S 3 × S 3 → SO(4)
given by (q1 , q2 ) → (u → q1 uq2−1 ) this corresponds to the element q → (q k , q −l )
in π3 (S 3 × S 3 ). Another way to describe these integers is as follows: If we
start with a principal S 3 × S 3 bundle Pk,l → S 4 , then we obtain two principal
S 3 bundles P/S 3 × 1 and P/1 × S 3 and these are now classified by their Euler
class −l and k.
To construct cohomogeneity one actions on these principal bundles, we
start with the group diagram
S3 × S3 × S3
(3.10)
0
(eip− θ , eiq− θ , eiθ ) ∪ (j, j, j)K−
0
(ejp+ θ , ejq+ θ , ejθ ) ∪ (i, i, i)K+
∆Q
which defines a cohomogeneity one manifold Pp10
as long as p± , q± ≡
− ,q− ,p+ ,q+
1 mod 4. S 3 × S 3 × 1 acts freely on it with quotient S 4 ; hence Pp− ,q− ,p+ ,q+ is
a principal S 3 × S 3 bundle over S 4 . As such, it is classified by two integers k
and l as above. The analogue of Proposition 3.3 is now
Proposition 3.11. The principal S 3 × S 3 bundle Pp− ,q− ,p+ ,q+ → S 4 is
2 − q 2 )/8. Hence every principal
classified by k = (p2− − p2+ )/8 and l = −(q−
+
3
3
4
S × S , respectively SO(4) bundle over S has a cohomogeneity one action by
G = S 3 × S 3 × S 3 / ± (1, 1, 1), respectively G = SO(4) × SO(3).
Proof. The formula for k and l follows from (3.3) since the group diagram
for P/S 3 × 1 × 1 is the S 3 × S 3 cohomogeneity one picture for Pq− ,q+ and hence
2 − q 2 )/8 and similarly k = (p2 − p2 )/8.
l = −(q−
+
−
+
350
KARSTEN GROVE AND WOLFGANG ZILLER
As before it follows that for each k, l, there exist solutions p± , q± to k =
2 − q 2 )/8 with p , q ≡ 1 mod 4. If k 6= 0, l 6= 0,
− p2+ )/8 and l = −(q−
± ±
+
there are only finitely many solutions, and for k 6= 0, l = 0, i.e., on Pk7 × S 3 ,
there exist infinitely many different cohomogeneity one actions.
The ineffective kernel of the S 3 × S 3 × S 3 action is ±(1, 1, 1); hence on
P ∗ = P/(−1, −1, 1) the effective action is by
(p2−
S 3 × S 3 × S 3 /h(−1, −1, −1), (−1, −1, 1)i = SO(4) × SO(3).
We finally point out that among the linear cohomogeneity one actions
on spheres [HL], only S 2 , S 3 , S 4 , S 5 and S 7 admit cohomogeneity one actions
where both singular orbits have codimension 2. Moreover in each case there
is only one effective action, and the groups are S 1 , T 2 , SO(3), SO(2)SO(3)
and SO(4) respectively. Among the nonlinear cohomogeneity one actions
on spheres, there exist infinitely many such actions by SO(2)SO(3) on S 5
(cf. Section 2 and [St2]).
We now explore the consequences to the existence of the SO(4) action on
S 7 for vector bundles and sphere bundles over S 7 . Since π6 (SO(k)) = 0 for
k = 2, 5, 6, 7 (see [Ja]) it follows that only principal SO(3) and SO(4) bundles
over S 7 can be nontrivial, and both admit two-fold covers to principal S 3
and S 3 × S 3 bundles. We first consider the case of principal S 3 bundles. As
was mentioned in the proof of (3.3), the cohomogeneity one picture for the
SO(4) action on S 7 is the same as that for P−3,1 . Hence, if we apply the
construction in Section 1 to obtain principal S 3 bundles over S 7 , one is forced
to consider the same cohomogeneity one picture as that for Pp− ,−3,p+ ,1 . By
Proposition 3.11, Pp− ,−3,p+ ,1 = Pk,1 is a principal S 3 × S 3 bundle over S 4 ,
where k = (p2− − p2+ )/8. One can of course also argue directly, that for every
principal S 3 × S 3 bundle Pk,1 over S 4 , we have Pk,1 /S 3 × 1 = P1 = S 7 and
hence Pk,1 can be regarded as a principal S 3 bundle over S 7 . As such, it is
classified by an element r ∈ π6 (S 3 ) = Z12 (see [Ja]), and it was shown in
[Ri3] that r = k(k + 1)/2 and hence each principal S 3 bundle over S 7 with
r = 0, 1, 3, 4, 6, 7, 9, 10 can be written in the form Pk,1 in infinitely many ways.
We thus obtain:
Corollary 3.12. Eight of the 12 principal S 3 bundles over S 7 , classified
by r = 0, 1, 3, 4, 6, 7, 9 and 10, admit infinitely many cohomogeneity one actions
by S 3 × S 3 × S 3 / ± (1, 1, 1).
As a consequence, the associated bundles over S 7 with fiber S 2 or R3 also
carry infinitely many metrics with nonnegative curvature. Note, however, as
was done in [Ri3], that the total space of the principal S 3 bundles over S 7 not
achieved by (3.12), i.e., r = 2, 5, 8, 11 are diffeomorphic to the corresponding
ones for r = 10, 7, 4, 1 ≡ −2, −5, −8, −11 mod 12. In fact, they are simply the
pull back of these bundles via the reflection R in the equator S 6 in S 7 . As a
CURVATURE AND SYMMETRY OF MILNOR SPHERES
351
consequence the corresponding associated S 2 and R3 bundles also have diffeomorphic total spaces. Thus all 3-dimensional vectorbundles and corresponding
sphere bundles over S 7 have complete metrics of nonnegative curvature.
Similarly, principal S 3 × S 3 bundles over S 7 are classified by (r, s) ∈ Z12
⊕ Z12 , and it follows as in (3.11) that every such bundle, with r, s = 0, 1, 3, 4, 6,
7, 9, 10, admits infinitely many cohomogeneity one actions by S 3 ×S 3 ×S 3 ×S 3 .
As before the principal bundles with r, s = 2, 5, 8, 11 as well as the corresponding associated bundles with fiber S 3 or R4 have total spaces diffeomorphic to
the ones with r, s = 0, 1, 3, 4, 6, 7, 9, 10. Also notice that the eight bundles with
(r, s) = (0, a) or (a, 0), a = 2, 5, 8, 11 clearly have nonnegative curvature since
the principal bundles are products of S 3 with principal S 3 bundles over S 7 . In
summary:
Corollary 3.13. All three-dimensional and 88 of the 144 four -dimensional vector bundles over S 7 , as well as the corresponding sphere bundles,
have metrics with nonnegative sectional curvature.
We conclude this section by pointing out that previously known methods
yield the following (see also [Ri4]):
Proposition 3.14. All vector bundles and all sphere bundles over S 5
admit complete metrics of nonnegative curvature.
Proof. Since π4 (SO(3)) = Z2 , π4 (SO(4)) = Z2 ⊕ Z2 , and π4 (SO(5)) = Z2
there are only 1,3, respectively 1 nontrivial vector bundle among the 3-, 4-,
respectively 5-dimensional vector bundles over S 5 . We will show that the total
space of each of the corresponding principal bundles is diffeomorphic to a Lie
group, such that the principal action is by isometries in the bi-invariant metric.
This implies that all vector bundles and sphere bundles over S 5 admit a metric
with nonnegative curvature.
The tangent bundle of S 5 gives rise to the nontrivial element in π4 (SO(5))
= Z2 and its principal bundle is SO(6) → SO(6)/SO(5) = S 5 .
As before, we can replace the principal SO(3) and SO(4) bundles with principal S 3 and S 3 × S 3 bundles respectively. The principal S 3 bundle SU(3) →
SU(3)/SU(2) = S 5 is the nontrivial element in π4 (S 3 ) = Z2 .
The nontrivial bundle corresponding to (1, 0) ∈ Z2 × Z2 = π4 (S 3 × S 3 ) is
given by the action of S 3 × S 3 = SU(2) × SU(2) on SU(3) × SU(2):
(α, β)(A, B) = (Aα−1 , Bβ −1 )
and similarly for (0, 1).
The action (α, β)(A, B) = (Aα−1 , αBβ −1 ) represents the nontrivial element (1, 1). Indeed, if we divide by the first SU(2) the map
SU(3) × SU(2)/SU(2) → SU(3) : (A, B) → A
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KARSTEN GROVE AND WOLFGANG ZILLER
is an SU(2)- equivariant diffeomorphism of SU(2) principal bundles, and if we
divide by the second SU(2), the map
SU(3) × SU(2)/SU(2) → SU(3) : (A, B) → AB
is an SU(2) equivariant diffeomorphism, and hence both bundles are nontrivial.
4. Almost free SO(3) actions
As we have seen in Section 3, there are typically many different ways
of representing the principal bundles discussed in this paper as cohomogeneity
one manifolds. This will in general yield different induced actions on associated
bundles, and will enable us, in particular, to prove Theorems C and D in the
introduction.
Recall, that any S 3 bundle over S 4 is associated to a principal SO(4)
bundle over S 4 , which in turn is determined by its two-fold universal cover, a
principal S 3 × S 3 bundle over S 4 . Each of these bundles are thus determined
by a pair of integers (k, l) ∈ Z × Z = π3 (S 3 × S 3 ) = π3 (SO(4)), where we
use the convention described in the previous section. For (k, l) ∈ Z × Z let
∗ → S4, P
4
3
Mk,l → S 4 , Pk,l
k,l → S denote the corresponding S bundle, principal
SO(4) bundle and principal S 3 × S 3 bundle respectively. In Section 3 we saw
that for any choice of integers p± , q± ≡ 1 mod 4, satisfying k = (p2− − p2+ )/8
2 − q 2 )/8 there is a cohomogeneity one action by S 3 × S 3 × S 3 on
and l = −(q−
+
Pk,l with diagram (3.10), which induces an effective action of SO(4)×SO(3) on
∗ . The SO(4) subaction is the free principal action on P ∗ and the subaction
Pk,l
k,l
by SO(3) is a lift of the cohomogeneity one action of SO(3) on S 4 . In particular
SO(3) acts on the total space of every associated S 3 bundle taking fibers to
fibers.
Theorem 4.1. The SO(3) action on Mk,l , induced from (3.10) as described above, preserves the S 3 fibration Mk,l → S 4 and has exactly the following orbit types:
(1), (Z2 ), (D2 ), (D |p− +q− | ), (D |p− −q− | ), (D |p+ +q+ | ), and (D |p+ −q+ | )
2
2
2
2
where D0 , in this context, should be interpreted as both SO(2) and O(2).
Proof. To compute the isotropy groups of this action, we apply the Isotropy
∗ ×
3
Lemma 1.7 to the corresponding ineffective S 3 action on Mk,l = Pk,l
SO(4) S =
3
3
3
3
Pk,l ×S 3 ×S 3 S . In the latter description S × S acts on S via quaternion
3
3
multiplication (Q1 , Q2 ) · v = Q1 vQ−1
2 . The isotropy groups of this S × S
3
3
3
3
action on S are 4S ⊂ S × S and conjugates thereof, i.e. the subgroups
Sa3 = {(b, aba−1 ) | b ∈ S 3 } for some a ∈ S 3 .
CURVATURE AND SYMMETRY OF MILNOR SPHERES
353
We can now read off the isotropy groups of the S 3 -action from (1.7). They
−1
3
3
3
3
are φ−1
± (Sa ) and φ0 (Sa ), where φ0 : Q → S × S is the diagonal embedding
3
3
and φ± : Pin (2) → S × S are the homomorphisms determined by φ− (eiθ ) =
(eip− θ , eiq− θ ), φ− (j) = (j, j) and φ+ (ejθ ) = (ejp+ θ , ejq+ θ ), φ+ (i) = (i, i).
3
iθ
jθ
kθ
Clearly φ−1
0 (Sa ) = h−1i = Z2 unless a ∈ he i, he i, he i, and in these
−1
3
cases φ0 (Sa ) = hii, hji, hki = Z4 , except when a = ±1, in which case
3
φ−1
0 (Sa ) = Q.
Now consider those eiθ1 , jeiθ2 ∈ Pin (2) such that (eip− θ1 , eiq− θ1 ) or
j(eip− θ2 , eiq− θ2 ) ∈ Sa3 . If a = eit , then aeip− θ1 a−1 = eiq− θ1 implies that
ei(p− −q− )θ1 = 1 and ajeip− θ2 a−1 = jeiq− θ2 implies that ei(p− −q− )θ2 −2it = 1.
Hence
3
2πi/(p− −q− )
, je2ti/(p− −q− ) i ⊂ Pin (2)
φ−1
− (Seit ) = he
for p− 6= q− . In the case of p− = q− , we get
3
iθ
1
φ−1
− (Seit ) = {e } = S ⊂ Pin (2)
3
if a = eit 6= ±1 and φ−1
− (S±1 ) = Pin (2).
If a = jeit , then aeip− θ1 a−1 = eiq− θ1 implies that ei(p− +q− )θ1 = 1 and
ip
aje − θ2 a−1 = jeiq− θ2 implies that ei(p− +q− )θ2 −2it = 1. Hence
3
2πi/(p− +q− )
, je2ti/(p− +q− ) i
φ−1
− (Sjeit ) = he
as the only possibility, since p− 6= −q− when p− , q− ≡ 1 mod 4.
If a 6∈ {eit } ∪ j{eit }, then the only θ1 with aeip− θ1 a−1 = eiq− θ1 is given by
θ1 = 0, π; i.e., eiθ1 = ±1. Moreover, for generic a, the equation ajeip− θ2 a−1 =
jeiq− θ2 has no solutions. For special values of a (depending on p− and q− )
there are precisely two values of θ2 ( θ2 and θ2 + π) with ajeip− θ2 a−1 = jeiq− θ2 .
3
Hence φ−1
− (Sa ) = Z2 or Z4
3
The groups φ−1
+ (Sa ) are computed in exactly the same way. Finally, to
obtain the isotropy groups of the effective action by SO(3), we only need
to observe that under the two-fold cover S 3 → SO(3) the images of Z4 , Q,
he2πi/p , je2ti/p i (for p even), {eiθ }, Pin (2) are equal to Z2 , D2 , Dp/2 , SO(2) and
O(2) respectively.
As pointed out in (3.7), for each (k, l) with k 6= 0, l 6= 0, there are
only finitely many solutions (p± , q± ) to the equations k = (p2− − p2+ )/8,
2 − q 2 )/8, when p , q ≡ 1 mod 4. As explained there also, one of
l = −(q−
± ±
+
these solutions can be written as (p− , p+ ) = (2k + 1, −2k + 1), (−k − 2, −k + 2)
or (k + 2, k − 2) when k ≡ 0 mod 2, k ≡ 1 mod 4 or k ≡ 3 mod 4 respectively. Similarly (q+ , q− ) = (2l + 1, −2l + 1), (−l − 2, −l + 2) or (l + 2, l − 2)
when l ≡ 0 mod 2, l ≡ 1 mod 4 or l ≡ 3 mod 4 respectively. If say l = 0,
(q− , q+ ) = (4n + 1, 4n + 1) is obviously a solution for all n.
We exhibit the isotropy groups, other than (1), (Z2 ) and D2 , of these
particular SO(3) actions on Mk,l in the following table:
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KARSTEN GROVE AND WOLFGANG ZILLER
k even
k odd
l even
D|k+l| , D|k−l±1|
D|k+2l±1|/2 , D|k−2l±3|/2
l odd
D|2k+l±1|/2 , D|2k−l±3|/2
D|k−l±4|/2 , D|k+l|/2
l = 0
D|2n+1±k| , D|2n±k|
D|4n+3±k|/2 , D|4n−1±k|/2
Table 4.2. Isotropy groups
In particular, we get:
Corollary 4.3. Each of the manifolds Mk,0 admit infinitely many inequivalent almost free SO(3) actions preserving the fibration Mk,0 → S 4 and
inducing the same cohomogeneity one action on S 4 .
Remark 4.4. Notice that Mk,0 are also precisely those Mk,l which can
be regarded not only as S 3 bundles over S 4 , but also as principal S 3 bundles.
Indeed, the glueing map q → {u → q k u} for the bundle Mk,0 commutes with
the right action by S 3 and hence S 3 acts freely on Mk,0 with quotient S 4 .
Thus Mk,0 = Pk and one can therefore also directly lift the SO(3) action on S 4
using the cohomogeneity one action on Pk = Pp− ,p+ from Corollary 3.7. But
this is an effective action of S 3 on Mk,0 , instead of SO(3), and one easily sees
that it also acts almost freely with isotropy groups the binary dihedral groups
he2πi/p± , ji. These actions of S 3 , finitely many for each k, commute with the
free principal action of S 3 , whereas the infinitely many almost free actions of
SO(3) in Corollary 4.3 do not.
A more detailed version of Corollary 4.3 in the special case of the Hopf
fibration S 7 = M1,0 → S 4 is included in the following result, which also implies
Theorem C in the introduction.
Theorem 4.5. For each n there is an action of SO(3) on S 7 which
preserves the Hopf fibration S 7 → S 4 and has exactly the following orbit types:
(1), (Z2 ), (D2 ), (D|2n−1| ), (D|2n| ), (D|2n+1| ), (D|2n+2| )
where as before (D0 ) stands for (SO(2)) and (O(2)). In particular, for n 6=
0, −1 this action is almost free. Moreover the action does not extend to the
disc D8 if n 6= 0, ±1, ±2.
Proof. The first part can just be read off from Theorem 4.1 and Table 4.2
when k = 1, l = 0. To prove that the actions do not extend to the disc, we use
the work of Oliver in [Ol] concerning the structure of fixed point free SO(3)
actions on discs. First, however, consider the case of an SO(3) action on D8
CURVATURE AND SYMMETRY OF MILNOR SPHERES
355
with nonempty fixed point set DSO(3) 6= ∅. Since SO(3) has only irreducible
representations in odd dimensions, it follows that the slice representation of
SO(3) at a fixed point, restricted to SO(2) ⊂ SO(3), has to have a fixed
vector and hence dim DSO(2) > 0. By Smith theory DSO(2) ⊃ DSO(3) has
the integral cohomology of a point (cf. e.g. [Br, Chap. III]). In particular any
component of DSO(2) with positive dimension has nonempty boundary, and
∂DSO(2) = DSO(2) ∩ ∂D = S SO(2) . Thus, if an almost free action of SO(3) on
S 7 extends to D8 , it cannot have fixed points.
Next, we will show, using [Ol], that any fixed point free action of SO(3) on
D8 has (Z3 ) or (D3 ) among its orbit types on the boundary sphere S 7 , which
then proves our theorem. From Corollary 1 of [Ol] we know in particular that
D3 occurs as isotropy group for any SO(3) action on D8 without fixed points.
In fact, from Lemmas 1 and 3 in [Ol], it follows that the octahedral group
O ⊂ SO(3) has an isolated fixed point in the interior of D8 and that D3 occurs
as an isotropy group of the linear representation of O at such a fixed point. In
particular, D3 is the isotropy of an interior point p ∈ D8 and dim DD3 > 0.
Again by Smith theory DZ3 ⊃ DD3 has the Z3 -cohomology of a point. Thus
each component of DZ3 intersects ∂D = S nontrivially in S Z3 .
Relative to an SO(3)-invariant metric on D8 , join p ∈ DD3 ⊂ DZ3 to a
closest point q ∈ ∂DZ3 = S Z3 inside DZ3 . In particular, Z3 ⊂ SO(3)q . But
SO(3)q also fixes the normal vector to the boundary at q, hence the minimal
geodesic above, and therefore p. Hence SO(3)q ⊂ SO(3)p = D3 , which implies
that SO(3)q = Z3 or D3 .
Remark 4.6. The group of linear symmetries of the Hopf fibration S 7 →
is given by (Sp(2)×Sp(1))/Z2 where Sp(2) acts via matrix multiplication on
7
S ⊂ H2 , and Sp(1) is the right Hopf action. The cohomogeneity one action
of SO(3) on S 4 defines an embedding of a maximal SO(3) in SO(5), which
under the two-fold cover Sp(2) → SO(5) lifts to a maximal Sp(1)m ⊂ Sp(2).
Hence (Sp(1)m ×Sp(1))/Z2 = SO(4) ⊂ (Sp(2)×Sp(1))/Z2 , which also happens
to be the cohomogeneity one action of SO(4) on S 7 with singular orbits of
codimension two, projects to SO(3) ⊂ SO(5) under the Hopf map. Thus there
are two linear lifts of the cohomogeneity one action of SO(3) on S 4 . One is
the almost free action by Sp(1)m on S 7 with isotropy groups 1 and Z3 , and
the other is the action by SO(3) = 4Sp(1)/Z2 ⊂ SO(4), which has isotropy
groups 1, Z2 , D2 , SO(2), and O(2). In particular, none of the actions in (4.5)
are linear, except n = 0, which is the latter one.
S4
We now analyze the ramifications of (4.1) to the Milnor spheres. Recall
that Milnor [Mi] showed that the Euler class of Mk,l → S 4 is equal to e = k + l.
Hence the Gysin sequence and Smale’s solution of the Poincaré conjecture
implies that Mk,l is homeomorphic to S 7 if and only if k +l = ±1. By changing
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KARSTEN GROVE AND WOLFGANG ZILLER
the orientation if necessary, we can assume that k + l = 1. For Theorems
A and D, we also need the diffeomorphism classification of these homotopy
spheres Mk,1−k due to Eells and Kuiper [EK]. According to [EK], Mk,1−k is
oriented diffeomorphic to Mm,1−m if and only if k(k − 1) ≡ m(m − 1) mod 56,
i.e. the oriented diffeomorphism class is given by k(k−1)
mod 28 in the group
2
Z28 of exotic 7-spheres. Notice also that a change in sign in Z28 corresponds
to a change in orientation of the manifold and hence the numbers 1 to 14
correspond to the 14 possible distinct diffeomorphism types of exotic spheres.
As was observed in [EK], k(k−1)
mod 28 takes on the following 16 val2
ues: 0,1,3,6,7,8,10,13,14,15,17,20,21,22,24,27 which via a change of orientation
corresponds to the numbers 0,1,3,4,6,7,8,10,11,13,14 and hence 11 of the 15
different diffeomorphism types of topological 7-spheres fiber over S 4 with S 3
as fiber. Using Theorems E and F, this completes the proof of Theorem A in
the introduction. In passing, we note that M2,−1 generates the group Z28 of
all homotopy 7-spheres via connected sum.
It is now clear that the SO(3) actions considered in (4.1) (cf. Table 4.2)
on Mk,1−k and Mm,1−m are, in general, different actions on the same homotopy sphere when k(k−1)
≡ m(m−1)
mod 28. To make this more concrete, we
2
2
exhibit the following special cases. As pointed out in [EK, p. 102], k(k − 1) ≡
m(m − 1) mod 56 if and only if m ≡ k or 1 − k mod 7 , and m ≡ k or 1 − k
mod 8 . Choosing the special case m ≡ k mod 56, we get:
Corollary 4.7. Let Mk,1−k be any of the homotopy 7-spheres considered
above. Then for each integer n, Mk,1−k supports an SO(3) action with the
following orbit types:
(1), (Z2 ), (D2 ), (D|k+56n+1±1|/2 ), (D|3(k+56n)−1±3|/2 )
if k is even, and
(1), (Z2 ), (D2 ), (D|k+56n−2±1|/2 ), (D|3(k+56n)−2±3|/2 )
if k is odd.
Of course, there are many more actions given by Theorem 4.1 on the
exotic spheres Mk,1−k , most of which are almost free. Even for the standard
sphere, we get many additional almost free actions, besides the ones described
in Theorem C, whenever Mk,1−k is diffeomorphic to S 7 , i.e. for k ≡ 0, 1 mod 7
and k ≡ 0, 1 mod 8. They preserve a different fibration of S 7 by 3-spheres, but
in this case, we get only finitely many actions for each fibration.
Not all of the actions in Table 4.2 and Corollary 4.7 are almost free.
Indeed, in the case of Mk,l with k = −l = 2r there exists no almost free lift
CURVATURE AND SYMMETRY OF MILNOR SPHERES
357
preserving the fibration, since (3.8) implies that the only action obtained from
(4.1) is the one described in Table 4.2. For the homotopy spheres Mk,1−k ,
the only actions in Table 4.2 which are not almost free occur in the case
of k = 0, 1, −2, 3. Of course for k = 0, 1, which corresponds to the Hopf
fibration, (4.5) gives rise to infinitely many almost free actions preserving the
fibration. For k = −2, 3, i.e. on M−2,3 = M3,−2 , (4.1) implies that there
exist one further action besides the one described in Table 4.2. It corresponds
to (p− , p+ ) = (5, 1), (q− , q+ ) = (−3, 5) and hence gives rise to an almost free
action with isotropy groups 1, Z2 , D2 , D3 , D4 . Thus in the case of the homotopy
spheres Mk,1−k there always exists at least one almost free action preserving
the fibration. This implies Theorem D in the introduction.
We also observe that the SO(3) actions on Mk,l extend to an action of
O(3). For this just note that the element −id ∈ SO(4) commutes with the
∗ and hence induces an action on
structure group and the SO(3) action on Pk,l
the associated bundle.
All the actions in this section on Mk,l are isometric actions with respect to
the nonnegatively curved metrics we constructed in Section 2. We now consider
the question whether these metrics can ever be isometric to each other, and
show that, at least in the case of the homotopy spheres Mk,1−k , this can almost
never be the case:
Proposition 4.8. If Mk,1−k and Mm,1−m are diffeomorphic, then the
metrics of nonnegative curvature constructed on them in Section 2 can only
be isometric, if the corresponding isometric SO(3) actions are conjugate. In
particular, we obtain infinitely many such metrics on each Mk,1−k which are
not isometric to each other.
Proof. To see this, we use the result by E. Straume that the degree of
symmetry of any exotic 7-sphere is at most 4, see [St1, Theorem C]. In other
words the dimension of any compact Lie group G that acts effectively on an
exotic 7-sphere is at most 4. Now fix a metric on Σ = Mk,1−k such that one of
the above SO(3) actions is isometric and let G ⊃ SO(3) be the id-component
of its full isometry group. Following Straume, G is either SO(3) or a finite
quotient of SO(3) × SO(2). In particular G contains only one subgroup SO(3),
so if one of the other actions of SO(3) on Σ is a subgroup of G, then the two
actions must be conjugate. Hence, if the SO(3) actions are inequivalent, the
corresponding metrics cannot be isometric and in fact have different isometry
groups.
We suspect that O(3) will always be the full isometry groups of the metrics we constructed on Mk,l ; see the next sections for some comments on this
question.
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KARSTEN GROVE AND WOLFGANG ZILLER
5. Remarks and open problems
Recall the two steps in our approach to the Cheeger-Gromoll problem:
(1) Any principal SO(k)-bundle over S 4 has a cohomogeneity one G-structure
with SO(k) ⊂ G (and with singular orbits of codimension two). (2) Any
cohomogeneity one G-manifold (with singular orbits of codimension two) admits a G-invariant metric with nonnegative curvature. As suggested in the
introduction, it is plausible that any cohomogeneity one manifold supports an
invariant metric with nonnegative curvature. This is just one of the reasons
for the following challenging:
Problem 5.1. Which principal SO(k)-bundles over S n with k ≤ n support
a cohomogeneity one G-structure with SO(k) ⊂ G?
Note that for each of the ways of writing S n as a cohomogeneity one
manifold (cf. [HL]), our construction in Section 1 will in general yield several
candidates for such bundles. In some cases it will give rise to infinitely many
such candidates, namely whenever S 1 is a normal subgroup of K− or K+ . One
can further increase the flexibility of our construction by using subactions of the
usual cohomogeneity one actions listed in [HL], which are still cohomogeneity
one (see [St2] for a complete list), and by making the actions ineffective. Of
particular interest here are of course SO(8)-bundles over S 8 , since 4095 exotic
15-spheres can be presented as (linear) 7-sphere bundles over the 8-sphere (cf.
[Sh] and [EK]).
In view of our examples, it would be interesting to study in more detail
the topology of the principal S 3 × S 3 bundles Pk,l → S 4 and their associated
3-sphere bundles Mk,l → S 4 . As we observed before, in the case of the sphere
bundles Mk,l , one can recover the Euler class e = k + l from the torsion in
H 4 . But for the principal bundles Pk,l the only nonzero cohomology groups
H ∗ (Pk,l , Z) are H 0 = H 3 = H 7 = H 10 = Z, H 4 = Z(k,l) . Indeed, in Section 3
we saw that among the total spaces Pk,1 there are at most seven diffeomorphism
types, and, using [HiR], it follows that there are precisely seven diffeomorphism
types. In the special case of the homotopy spheres Mk,1−k , the total space has
been classified up to diffeomorphism in [EK]. It would be interesting to extend
this classification:
Problem 5.2. Classify the manifolds Pk,l and Mk,l up to homotopy,
homeomorphism and diffeomorphism type.
See [JW] and [Tam] for a partial classification of Mk,l up to homotopy
and homeomorphism type. For general results about 2-connected 7-manifolds
see [Wa] and [Wi]. Also notice that it was shown in [DW] that for the corresponding vector bundles Ek,l → S 4 , the total spaces are diffeomorphic if and
only if they are isomorphic as vector bundles.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
359
Since our manifolds Mk,l have trivial π2 and π3 = Zk+l , one easily sees
that if one of the manifolds Mk,l is homotopy equivalent to a homogeneous
space G/H, then G/H = Sp(2)/Sp(1), where Sp(1) is one of the three possible
embeddings of Sp(1) in Sp(2). Hence G/H = Sp(2)/Sp(1) × 1 = S 7 , G/H =
Sp(2)/4Sp(1) = T1 S 4 or G/H = Sp(2)/Sp(1) = SO(5)/SO(3) where SO(3) ⊂
SO(5) is the maximal embedding given by the cohomogeneity one action of
SO(3) on S 4 . It was shown in [Be] that B 7 = SO(5)/SO(3) carries a metric
of positive sectional curvature and that H 4 (B 7 , Z) = Z10 . One easily shows
that the first Pontryagin class of the tangent bundle of B 7 is equal to 6 times
a generator in H 4 = Z10 . Furthermore, in [Tam] it was shown that the first
Pontryagin class of the tangent bundle of Mk,l is equal to ±4l times a generator
in H 4 (Mk,l , Z) = Zk+l . Hence B 7 cannot be homeomorphic to a principal S 3
bundle over S 4 ; i.e. l = 0. But it would be interesting to know if it can be
homeomorphic or diffeomorphic to a sphere bundle.
For the principal S 3 bundles Pk over S 4 , we have that Pk is diffeomorphic
to P−k and that P±1 = S 7 . Furthermore, as was observed in [Ri2], P±2 = T1 S 4
since on T1 S 4 = Sp(2)/4Sp(1) one has the free action by S 3 given by left multiplication with diag (q, 1) and since H 4 (T1 S 4 , Z) = Z2 , it follows that this
principal S 3 bundle has k = ±2. Hence P±1 and P±2 are diffeomorphic to
homogeneous spaces. From the above, it follows that all other Pk are strongly
inhomogeneous; i.e. they do not have the homotopy type of a homogeneous
space, except possibly P±10 which is at least not homeomorphic to a homogeneous space.
For the associated 2-sphere bundles Mk → S 4 considered in (3.9), it follows
from [On, Theorem 6] that the only homogeneous spaces that have the same
integral cohomology groups as CP 3 , are CP 3 = M±1 itself and S 2 × S 4 = M0 .
Hence Mk with |k| ≥ 2 do not have the homotopy type of a homogeneous
space.
By construction, the total space P , respectively M , of any principal bundle, respectively associated sphere or vector bundle considered in this paper,
as well as the base S 4 , is the union D− ∪ D+ of two disc bundles with common
boundary S = ∂D− = ∂D+ . Moreover, relative to the metrics of nonnegative
curvature on D− ∪ D+ , S is totally geodesic. From the Cheeger-Gromoll soulconstruction [CG] and Perelman’s rigidity theorem [Pe] it follows in particular
that there are two-planes with zero curvature at every point of D− ∪ D+ .
All cohomogeneity one G-manifolds considered in this paper have
G = S 3 , S 3 × S 3 or S 3 × S 3 × S 3 (acting possibly ineffectively). For the metrics
on D± = G ×K± D2 , note that we can choose a fixed bi-invariant metric on G
scaled by a fixed constant (e.g. a = 4/3) in the K± direction, and on D2 we
choose a metric dt2 +f 2 (t)dθ2 with a fixed convex function f . Although K± and
hence Qa depends on the particular example, it easily follows from (2.4) and
360
KARSTEN GROVE AND WOLFGANG ZILLER
the Gray-O’Neill submersion formula that there is a uniform bound C for the
curvatures of all principal bundles considered here; i.e. 0 ≤ sec(P ) ≤ C. But,
as explained in (2.7), there is no bound on the diameter, diam(P ), since the
length of the circles K± goes to infinity. It is also apparent that all examples
have a uniform lower bound on their volumes; i.e., there is a v > 0 such that
vol (D− ∪ D+ ) ≥ v since this is true of vol (S). Similar bounds for curvature,
volume and diameter hold for the associated bundles and the base S 4 .
All these examples complement Cheeger’s classical finiteness theorem [Ch2]
and recent finiteness theorems by Petrunin-Tuschmann [PT] and Tapp [Ta].
Since our examples of S 3 bundles over S 4 are 2-connected, they illustrate
the sharpness of the following, even within the class of nonnegatively curved
manifolds.
Theorem 5.3 (Petrunin-Tuschmann). For each n and D, C > 0 there
exist only finitely many diffeomorphism types of simply connected compact Riemannian n-dimensional manifolds M , with |sec M | ≤ C, diam M ≤ D and
finite π2 (M ).
In the special case where the lower curvature bound is a fixed positive
number δ > 0, the same conclusion was obtained simultaneously by Fang and
Rong in [FR] (in that case the bound on diam M is automatic by the BonnetMyers theorem). Motivated by this, the following conjecture was proposed in
[FR]:
Conjecture (Rong).
For each n, there are at most finitely many
2-connected positively curved n-manifolds.
Our examples show that this conjecture is false, if we replace positive with
nonnegative curvature.
The following result was first obtained in [GW] in the special case where
Σ = S n (1). It was then extended to arbitrary souls in [Ta].
Proposition 5.4. For each n and C, D, V > 0 and each metric on Σ
with diam Σ ≤ D and vol Σ > V , there exist only finitely many vector bundles
M over Σ with a complete metric of nonnegative curvature, such that Σ is the
soul and such that the sectional curvatures of all 2-planes of M either tangent
to Σ or normal to Σ are bounded above by C.
By the above remarks, in our examples of 3- and 4-dimensional vector
bundles over S 4 , we have bounds on the curvatures of M and the volume of
the soul Σ, which is always the zero section of the vector bundle and isometric
to the metric on the base S 4 . But the diameter of the base necessarily goes to
infinity since the length of the circles K± /H = S 1 goes to infinity.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
361
We conclude our discussion with a few more remarks about the geometry
and symmetry of our examples, in particular the principal S 3 × S 3 bundles
Pk,l → S 4 and their associated sphere bundles S 3 → Mk,l → S 4 and vector
bundles R4 → Ek,l → S 4 . Much work has been done previously on trying to
construct metrics with nonnegative or positive curvature on the total spaces
Pk,l , Mk,l and Ek,l . A natural approach is to consider Kaluza-Klein type metrics
on the principal L bundle P , where one chooses a principal connection to define
the horizontal space, pulls back the metric from the base to the horizontal
space, and defines the metric on the fiber to be a bi-invariant or left-invariant
metric on L. This metric then also induces a metric on the associated sphere
bundles and vector bundles. We call these metrics connection-type metrics. In
all three cases, the fibers of the projection onto the base are totally geodesic
and isometric to each other. The metrics of positive Ricci curvature on Pk,l
and Mk,l constructed in [Na] and [Po] are exactly of this type. But for the
construction of nonnegatively curved metrics this approach has been successful
only in the case where the principal bundle is a Lie group or a homogeneous
space. In [DR] it was shown that the only case in which the induced metric
on Mk,l has positive curvature, is when k = 0, l = ±1 or k = ±1, l = 0, i.e.
when Mk,l = S 7 . It would be interesting to know if nonnegatively curved
connection-type metrics exist on the bundles Mk,l with kl 6= 0, 1. The metrics
in our examples are not of this type. In our case, we will show that the metrics
on the S 3 fibers (as well as on the base S 4 ) are cohomogeneity one metrics,
and that the fibers are not totally geodesic.
On Pk,l we have the cohomogeneity one action by G = S13 × S23 × S33 with
the principal bundle action given by S13 × S23 (but acting on the left on P ). By
construction, the projection Pk,l → S 4 is a Riemannian submersion, with the
horizontal distribution given by a principal connection, since the metric is S13 ×
S23 invariant. The same follows for the associated bundles Mk,l = Pk,l ×S 3 ×S 3 S 3
1
2
and Ek,l = Pk,l ×S 3 ×S 3 R4 . In all three cases, the metric on the base is given
1
2
by the submersed metric on Pk,l /S13 × S23 , and as a cohomogeneity one metric
on S 4 under the action of S33 , is described by three functions f1 (t), f2 (t), f3 (t),
the length of the three action fields i∗ , j ∗ , k ∗ along c(t) with i, j, k in the Lie
algebra of S33 . Here c(t) is a fixed geodesic perpendicular to all orbits, as in
Section 1. Invariance of the metric under the isotropy action of H = Q implies
that these action fields must be orthogonal. It follows from Perelman’s rigidity
theorem that two of these functions are equal to 1, f2 (t) = f3 (t) = 1 on D−
and f1 (t) = f3 (t) = 1 on D+ .
The metric on the fiber of Pk,l → S 4 over the point c(t) in S 4 is given by
a left-invariant metric Qt on S13 × S23 . But this metric depends on t, and hence
the fibers are not totally geodesic. This completely describes the metric on
Pk,l . It is interesting to observe that our metrics are just slightly more general
362
KARSTEN GROVE AND WOLFGANG ZILLER
than connection-type metrics in that the metrics on the fiber are allowed to
depend on a single parameter t.
Notice that the left-invariant metric Qa on G is also right invariant under
0 which hence acts S 1
the maximal torus T±3 ⊂ S13 × S23 × S33 containing K±
ineffectively and by isometries on each half D± = G ×K± D2 of Pk,l . But the
intersection of T− and T+ acts trivially. Furthermore, the first two components
of T±3 act by isometries via right translation on the left-invariant metric Qt on
S13 × S23 .
We now consider the geometry of the associated bundles Mk,l = Pk,l ×S 3 ×S 3
1
2
3
S (r) and Ek,l = Pk,l ×S 3 ×S 3 R4 , where we also allow ourselves the freedom
1
2
of varying the radius in S 3 (r). The horizontal distribution and the metric on
the base S 4 is the same as before, and we only need to describe the metric on
the fibers. The fiber of the S 3 bundle Mk,l → S 4 , over the point c(t) ∈ S 4 can
be described as S13 × S23 ×S 3 ×S 3 S 3 (r) = S 3 where the metric on S13 × S23 is
1
2
given by the left-invariant metric Qt . Only the right translations on S13 × S23 ,
that are still isometries of Qt , are isometries of this metric on S 3 . These right
translations consist of the action by T 2 = (eiθ , eiψ ) (in the case of D− ) and this
action of T 2 on S 3 is the standard cohomogeneity one action on S 3 . Hence all
fibers of Mk,l → S 4 are cohomogeneity one metrics on S 3 (and in particular not
homogeneous). Choosing a basis of T 2 , the metric on S 3 can be described by
the length and inner product of the corresponding action fields along a normal
geodesic in the fiber. But, unlike in the case of S 4 , since the principal isotropy
group of the T 2 action is trivial, the inner product between these two action
fields does not have to be 0. Hence the metric on the fiber is described by three
functions h1 (s, t), h2 (s, t), h3 (s, t), where s is the arc length parameter of a normal geodesic in the cohomogeneity one metric on the fiber S 3 over c(t). Thus,
the metric on Mk,l is completely described by (f1 , f2 , f3 , h1 , h2 , h3 ): I ×I → R6 .
Similarly, the metric on the fibers of Ek,l → S 4 are cohomogeneity two
metrics dt2 +gt dθ2 with gt a cohomogeneity one metric on S 3 as above. In both
cases, the fibers again change from point to point and hence are not totally
geodesic.
Notice that on Mk,l (but not on Ek,l ) one can describe a different metric
using the identification Pk,l ×S 3 ×S 3 S 3 = Pk,l /4S 3 as a submersed metric from
1
2
Pk,l . For this metric, the horizontal distribution and the metric on the base is
the same, but the metric on the fibers is now the metric on S 3 = 4S 3 S13 × S23
induced from the left-invariant metric Qt on S13 × S23 which is invariant under
right translations by T 2 and hence again only a cohomogeneity one metric on
S 3 . To compare this metric with the previous metric, if we consider the metric
on Mk,l = Pk,l ×S 3 ×S 3 S 3 (r) and let r go to infinity, then the limit is the new
1
2
metric just described. Indeed, if we set L = S13 × S23 , H = 4S 3 , then one
has the identification P ×L L/H ' P/H given by [(p, `H)] → H · `−1 p. The
CURVATURE AND SYMMETRY OF MILNOR SPHERES
363
fiber L ×L L/H then gets identified with H L and if Qt (X, Y ) = Q(At X, Y )
it follows as in (2.1) that the metric is induced from the left invariant metric
2A
t
Q( Art +r
2 X, Y ), which as r goes to infinity, converges to Qt .
Next, we consider the isometry group of our examples. As was explained
in Section 4, the action of SO(3) on the principal bundle Pk,l descends to an
action of SO(3) on the total space of Mk,l which acts by isometries in the
metric of nonnegative curvature that we constructed. Furthermore, the action
can be extended to an isometric action by O(3). We suspect that this group
will always be the full isometry group of our metrics. In the special case of
exotic spheres Mk,1−k , it follows from [St1, Theorem C], that the id component
of the full isometry group can be at most SO(3) × SO(2) or one of its finite
quotients. An affirmative answer to the following question would of course rule
out such an extension, and would imply that the group SO(3) is always the
id-component of the full isometry group.
Problem 5.5. Does the SO(3) subaction of an (almost) effective SO(3) ×
SO(2) action on an (exotic) 7-sphere have isotropy groups containing SO(2)?
In [Da] it was observed that there are natural lifts of the cohomogeneity
one action of SO(3)SO(2) on S 4 to each of the manifolds Mk,l . Also, each
exotic 7-sphere can be exhibited as a Brieskorn variety, and as such it again
supports a natural action of SO(3)SO(2). If the exotic sphere is of the form
Mk,1−k , this action is in general different from the previous ones. In either
case, however, the subaction of SO(3) is never almost free.
We also remark, that the action of SO(3)SO(2) on S 4 lifts not only to
Mk,l as in [Da], but to the principal bundle Pk,l as well. But this lift does
not commute with the free action of S 3 × S 3 and hence one does not obtain a
cohomogeneity one action on Pk,l as we do in our examples.
An essential difference between our examples and the Gromoll-Meyer metric [GM] on M2,−1 , is that it has a 4-dimensional isometry group, which agrees
with the action of SO(3)SO(2) in [Da]. We finally rephrase the description of
this metric on the Gromoll-Meyer sphere in our context, thereby exhibiting
similarities and differences.
Consider the following subgroup G = (S 1 ×S13 )×(S23 ×S33 ) ⊂ Sp(2)×Sp(2):
(ÃÃ
cos θ − sin θ
sin θ
cos θ
!
!
· diag (q1 , q1 ), diag (q2 , q3 )
)
¯
¯
¯ qi ∈ Sp(1) .
This subgroup acts on Sp(2) (via left and right multiplication) and any two
combinations of the S 3 factors act freely and one easily sees that in all cases
the quotient is S 4 on which S 1 × S 3 (the S 3 being the remaining S 3 factor)
acts by cohomogeneity one with the standard sum action. Hence G also acts
364
KARSTEN GROVE AND WOLFGANG ZILLER
by cohomogeneity one on Sp(2) with singular orbits of codimension two and
three, and isometrically with respect to the bi-invariant metric with nonnegative curvature. If one chooses the free action by S23 × S33 , the principal S 3 × S 3
bundle over S 4 is the two-fold cover of the frame bundle of the tangent bundle
of S 4 and hence P1,1 = Sp(2). If one chooses the free action by S13 × S23 , it
was shown in [GM] that one obtains the principal bundle P2,−1 and hence the
associated sphere bundle P2,−1 ×S 3 ×S 3 S 3 = P2,−1 /41,2 S 3 is the exotic sphere
1
2
M2,−1 with a submersed metric of nonnegative curvature. As in our case, the
action of S 1 ×S33 descends to an action of the associated S 3 bundle M2,−1 which
becomes an effective action by SO(3)SO(2) and which, by [St1, Theorem C], is
the id-component of the isometry group of the Gromoll-Meyer metric. Notice
that, as in our case, we also get a family of metrics with nonnegative curvature
on the Gromoll-Meyer sphere by considering Sp(2) ×S 3 ×S 3 S 3 (r), and as r goes
1
2
to infinity we obtain the Gromoll-Meyer metric in the limit.
Of course, as a consequence of our results, it follows that P2,−1 = Sp(2)
also has infinitely many cohomogeneity one actions by S 3 × S 3 × S 3 , but with
singular orbits of codimension two. Another major difference between our
metrics induced by these actions and the Gromoll-Meyer metric, is that in
their example there exists an open set of points in M2,−1 on which every twoplane has positive curvature, whereas in our example, by construction, there
are always 2-planes of 0 curvature at every point.
Motivated by Proposition 4.8 and Theorem G we conclude our discussion
with the following natural question.
Problem 5.6. On each of the Milnor spheres (including the standard
sphere), as well as on the homotopy RP 5 in Theorem G, does the space of
metrics with nonnegative sectional curvature have infinitely many components?
In a similar vein, in [KS] it was shown that on some of the homogeneous
spaces SU(3)/S 1 the space of metrics with positive sectional curvature has at
least two components.
Added in proof. Motivated by Problem 5.2, D. Crowley and C. Escher
[CE] recently completed the classification of the total spaces Mk,l of S 3 bundles over S 4 up to orientation preserving and reversing homotopy equivalence,
homeomorphism and diffeomorphism.
In [KiS] N. Kichloo and K. Shankar partially answered another question raised in Section 5 by showing that the positively curved Berger space
B 7 = SO(5)/SO(3) is PL-homeomorphic to an S 3 bundle over S 4 . It remains
to compute the Eells-Kuiper invariant for this space in order to see if it is
diffeomorphic to one.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
365
It also follows from [CE] that all Mk,l with k + l = 10 are homotopy
equivalent to each other. Hence for the total spaces Pk of principal S 3 bundles
over S 4 , it follows that P10 is homotopy equivalent to the homogeneous space
B 7 , whereas, as we observed in Section 5, all other Pk , k ≥ 3 are strongly
inhomogeneous.
University of Maryland, College Park, MD
E-mail address: kng@math.umd.edu
University of Pennsylvania, Philadelphia, PA
E-mail address: wziller@math.upenn.edu
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